On Reducible Degeneration of Hyperelliptic Curves and Soliton Solutions

TL;DR

This paper investigates how hyperelliptic curves degenerate into reducible forms and how these degenerations relate to the emergence of various soliton solutions in the KP-hierarchy, including both regular and singular types.

Contribution

It demonstrates the existence of soliton solutions as limits of hyperelliptic solutions using the Sato Grassmannian and clarifies the relation between singularities and solution matrices.

Findings

Hyperelliptic solutions converge to soliton solutions in degenerations.

Regular soliton solutions correspond to reducible rational curves.

Singularities in solutions are reflected in the structure of solution matrices.

Abstract

In this paper we consider a reducible degeneration of a hyperelliptic curve of genus $g$. Using the Sato Grassmannian we show that the limits of hyperelliptic solutions of the KP-hierarchy exist and become soliton solutions of various types. We recover some results of Abenda who studied regular soliton solutions corresponding to a reducible rational curve obtained as a degeneration of a hyperelliptic curve. We study singular soliton solutions as well and clarify how the singularity structure of solutions is reflected in the matrices which determine soliton solutions.